3.497 \(\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=390 \[ -\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{4 a^3 \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (c^2-d^2\right ) \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f} \]

[Out]

(-4*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) - (4*a^3*(
4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*a^3*(c - 5*d)*Cos[e + f*x]
*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))
/(9*d*f) + (4*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(
c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^3 -
27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d
)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.775134, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{4 a^3 \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (c^2-d^2\right ) \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(-4*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) - (4*a^3*(
4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*a^3*(c - 5*d)*Cos[e + f*x]
*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))
/(9*d*f) + (4*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(
c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^3 -
27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d
)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{2 \int (a+a \sin (e+f x)) \left (a^2 (c+7 d)-2 a^2 (c-5 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2} \, dx}{9 d}\\ &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{2 \int (c+d \sin (e+f x))^{3/2} \left (a^3 (c+7 d)+\left (-2 a^3 (c-5 d)+a^3 (c+7 d)\right ) \sin (e+f x)-2 a^3 (c-5 d) \sin ^2(e+f x)\right ) \, dx}{9 d}\\ &=\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{4 \int (c+d \sin (e+f x))^{3/2} \left (-\frac{3}{2} a^3 (c-33 d) d+\frac{1}{2} a^3 \left (4 c^2-27 c d+119 d^2\right ) \sin (e+f x)\right ) \, dx}{63 d^2}\\ &=-\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{8 \int \sqrt{c+d \sin (e+f x)} \left (-\frac{3}{4} a^3 d \left (c^2-138 c d-119 d^2\right )+\frac{3}{4} a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sin (e+f x)\right ) \, dx}{315 d^2}\\ &=-\frac{4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{16 \int \frac{\frac{3}{8} a^3 d \left (c^3+387 c^2 d+471 c d^2+165 d^3\right )+\frac{3}{8} a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{945 d^2}\\ &=-\frac{4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}-\frac{\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{315 d^3}+\frac{\left (2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{315 d^3}\\ &=-\frac{4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{\left (2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{315 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{315 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{4 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{315 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.30345, size = 318, normalized size = 0.82 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (d (c+d \sin (e+f x)) \left (\left (-216 c^2 d+32 c^3-3828 c d^2-2910 d^3\right ) \cos (e+f x)+2 d \left (5 d (10 c+27 d) \cos (3 (e+f x))-\sin (2 (e+f x)) \left (6 c^2+432 c d-35 d^2 \cos (2 (e+f x))+511 d^2\right )\right )\right )-16 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (387 c^2 d+c^3+471 c d^2+165 d^3\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )\right )}{1260 d^3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(a^3*(1 + Sin[e + f*x])^3*(-16*(d^2*(c^3 + 387*c^2*d + 471*c*d^2 + 165*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (
2*d)/(c + d)] + (4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/
4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d
*(c + d*Sin[e + f*x])*((32*c^3 - 216*c^2*d - 3828*c*d^2 - 2910*d^3)*Cos[e + f*x] + 2*d*(5*d*(10*c + 27*d)*Cos[
3*(e + f*x)] - (6*c^2 + 432*c*d + 511*d^2 - 35*d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))))/(1260*d^3*f*(Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Maple [B]  time = 1.2, size = 1613, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x)

[Out]

2/315*a^3*(-1044*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/
2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^6-8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+si
n(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d)
)^(1/2))*c^6+714*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/
2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^6+4*c^4*d^2+8*((c+d*sin(f*x+e))/(c-d))^(1/2
)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))*c^5*d+c^3*d^3*sin(f*x+e)-243*c^2*d^4*sin(f*x+e)-704*c*d^5*sin(f*x+e)+85*c*d^5*sin(f*x+e)^5+
53*c^2*d^4*sin(f*x+e)^4+351*c*d^5*sin(f*x+e)^4-c^3*d^3*sin(f*x+e)^3+243*c^2*d^4*sin(f*x+e)^3+619*c*d^5*sin(f*x
+e)^3-4*c^4*d^2*sin(f*x+e)^2+27*c^3*d^3*sin(f*x+e)^2+413*c^2*d^4*sin(f*x+e)^2-21*c*d^5*sin(f*x+e)^2-60*((c+d*s
in(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f
*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d^2+1048*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d)
)^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^3+
1104*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^4-1056*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x
+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/
2))*c*d^5+54*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*E
llipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^5*d-214*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+si
n(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d)
)^(1/2))*c^4*d^2-1212*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)
)^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^3-492*((c+d*sin(f*x+e))/(c-d))^(1/
2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),(
(c-d)/(c+d))^(1/2))*c^2*d^4+1158*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*
x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^5-466*c^2*d^4+135*d^6*sin
(f*x+e)^5+203*d^6*sin(f*x+e)^4+195*d^6*sin(f*x+e)^3-238*d^6*sin(f*x+e)^2-330*d^6*sin(f*x+e)+35*d^6*sin(f*x+e)^
6-27*c^3*d^3-330*c*d^5)/d^4/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{3} d \cos \left (f x + e\right )^{4} + 4 \, a^{3} c + 4 \, a^{3} d -{\left (3 \, a^{3} c + 5 \, a^{3} d\right )} \cos \left (f x + e\right )^{2} +{\left (4 \, a^{3} c + 4 \, a^{3} d -{\left (a^{3} c + 3 \, a^{3} d\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral((a^3*d*cos(f*x + e)^4 + 4*a^3*c + 4*a^3*d - (3*a^3*c + 5*a^3*d)*cos(f*x + e)^2 + (4*a^3*c + 4*a^3*d -
(a^3*c + 3*a^3*d)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3*(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(3/2), x)